Page 64 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.6 Tunneling 55

⠇ iá
C ˆ F e (iáÿâ)a
2â
(2:56)

â ÿ iá
D ˆ F e (iá‡â)a
2â
Application of equation (2.54) to (2.47 a) and (2.50) gives
A ‡ B ˆ C ‡ D
(2:57)
iá(A ÿ B) ˆ â(C ÿ D)
Elimination of B from the pair of equations (2.57) and substitution of equations
(2.56) for C and for D yield
1
A ˆ [(⠇ iá)C ÿ (â ÿ iá)D]
2iá
e iáa
2 ÿâa
2 âa
ˆ F [(⠇ iá) e ÿ (â ÿ iá) e ]
4iáâ
2
At this point it is easier to form jAj before any further algebraic simpli®ca-
tions
2

jAj ˆ A A
1
2 2 2 2 ÿ2âa 2 2 2 2âa 4 4
ˆjFj [(⠇ á ) e ‡ (⠇ á ) e ÿ (â ÿ iá) ÿ (⠇ iá) ]
16á â
2 2
1
2 2 2 2 âa ÿâa 2 2 2
ˆjFj [(á ‡ â ) (e ÿ e ) ‡ 16á â ]
16á â
2 2
" #
2
2 2
(á ‡ â )
2 2
ˆjFj 1 ‡ sinh âa
4á â
2 2
where equation (A.46) has been used. Combining this result with equations
(2.48), (2.51), and (2.52), we obtain
" # ÿ1
V 2 p 
2
T ˆ 1 ‡ 0 sinh ( 2m(V 0 ÿ E) a=") (2:58)
4E(V 0 ÿ E)
To ®nd the re¯ection coef®cient R, we eliminate A from the pair of
equations (2.57) and substitute equations (2.56) for C and for D to obtain
1 e iáa 2 2 âa 2 2 ÿâa
B ˆ [ÿ(â ÿ iá)C ‡ (⠇ iá)D] ˆ F [(á ‡ â )e ÿ (á ‡ â )e ]
2iá 4iáâ
e iáa
2
2
ˆ F (á ‡ â )sinh âa
2iáâ